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The gradient of the potential function satisfies Equation (7). On an Eulerian grid with mesh size h, the partial differential equation can be discretized as 4φi, j − φi+1, j − φi−1, j − φi, j −1 − φi, j +1 ρi, j = , 2 h 0 which leads to a set of linear equations, denoted by A = ρ, where is a vector of unknown potentials and ρ is a vector of (scaled) charge densities. The matrix A is large, sparse, symmetric, and positive definite. The complexity of most matrix solvers depends on the size of A. For the simulation of auroral dynamics, a fine grid is required to resolve the layer structure, especially when the formation of surges is simulated.

Figure 8 shows the coefficients of the vertices that reproduce the quadratic polynomials x 2 , xy, and y 2 over the triangular and quadrilateral portions of the mesh. Notice that the coefficients for y 2 do not agree at the boundary, so S cannot possibly be C 2 at the boundary. ACM Transactions on Graphics, Vol. 24, No. 1, January 2005. On C 2 Triangle/Quad Subdivision • 33 Our goal is to construct a new subdivision scheme Sˆ such that Sˆ satisfies equation 1 for i = 1 . . 5, ˆ = 1 v where v is a new eigenvector corresponding to y 2 .

FD’s are the most commonly applied to the PDE formulation of Maxwell’s equations in the time domain [Anderson 2001]. This method is based on the straight discretization of the differential equations, and ¨ ¨ it is suitable for modeling complex media [Karkk ainen 2002]. This motivated its use in the simulation algorithms described in this article. In order to accelerate the convergence of the iterative solutions of the PDEs, we used a multigrid method. Recently, Aruliah and Asher [2002] also used a multigrid scheme to speed-up the simulation of three-dimensional electromagnetic problems in geophysical regimes.

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ACM Transactions on Graphics (January) by Association for Computing Machinery

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